# Deontic Logic

Updated on January 26, 2015

## Deontic Modalities

In ordinary language, certain qualifying phrases (often adverbial), generate what are called refentially opaque contexts: this is a technical issue but one that has a bearing on what the appropriate logical analysis for such phrases is. Deontic phrases like "it is morally obligatory that___" or "it is morally permissible that___" or "it morally forbidden that____" and so on, create referentially opaque contexts. Their characteristic logic cannot be the elementary logic of propositions for a reason we will briefly explain.

The basic logics - propositional and its predicate extension - are defined, one could say, by the property of extensionality: this principle, also known as Leibniz's Extensionality Principle, states: if either one of two logically equivalent propositions is substituted for the other within a compound proposition, then logical meaning is preserved. The broader version allows for preservation of truth value when co-referring terms are substituted for each other in expressions like F(a). Logical meaning, extensionally approached, is a matter of truth value (whether a proposition is true or false.) Indeed, difficult as it might be to penetrate behind the Leibniz principle when you first hear about it, the point is that "truth is preserved" when any other proposition with the same truth value (true OR false) is substituted for a given proposition. The substitution obtains salva veritatis as we say.

This is surprising to the uninitiated. How can extensional logics be taking determination of truth value (true or false) to be unaffected when some irrelevant, unrelated other proposition that happens to have the same truth value is substituted in? The answer to this is that deductive properties like argument-validity, consistency of collections of statements, and whether a statement makes a trivial point or not - none of them is affected by the more robust determination of meaning that would take into account the content of what the proposition is about. Conversely, if context matters for determination of logical meaning, we are not dealing with a basic or extensional logic.

Consider an example: assume that, in a given model we draw on given information, "John is lazy" is given as true. "Jill is lazy" is given as false. Therefore, for this model, "John is lazy" and "It is not the case that Jill is lazy" are logically equivalent with each other (both being true.) If you have the proposition "John is lazy and Jill is lazy", which is false, and you substitute "It is not the case that Jill is lazy and Jill is lazy" the truth value - still false - does not change. Now, notice a difference. The latter conjunction - "It is not the case that Jill is lazy and Jill is lazy" - has the logical form of a contradiction: so, it would have to be false in ANY model we can possibly have given to us. It is a logical falsehood. But the other conjunction - "John is lazy and Jill is lazy" - cannot be false in any model: nothing prevents us from drawing a different model in which this statement is true because, in that model, both John and Jill are given as belonging to the extension of the predicate "lazy." This shows that our extensionality principle, which gives us an elegantly simple logic, does not wreak mischief: it will not make us accept as logical truths or as logical falsehoods statements that are simply contingently true and contingently false. The crucial deductive properties are not affected by the stipulation that we can substitute logical equivalents for each other within extensional contexts.

In an intensional context this does not work. Deontic phrases generate what are called referentially opaque contexts. Let us see how. Take the proposition "it should not be the case, morally speaking, that recreational killing is legal." It so happens that "recreational killing is legal" is false in most relevant contexts (or so one hopes.) The whole modal proposition, "it should not be the case that recreational killing is legal" is true, let's say. Now we can easily find some other false proposition which we can plug into "it should not be the case that ---" so that what fills in the dots is also false but the whole comes out as true on some view. For instance it is also false that "it is legal for a starving person to steal food" but we might posit that "it should not be the case that it is legal for a starving person to steal food" ought to be taken as true. It does not matter that views on such matters vary: it is sufficient that we can find such contexts, open-endedly, to show that substitution of false for false, or of true for true, does NOT guarantee that the truth value of the whole modal proposition is preserved. Deontic phrases (like "should" and "should not", and "must" and "may be permitted" and "it is forbidden", etc.) generate contexts for which substitution of equivalents fails. Whatever a Deontic Logic is, it cannot be an extensional system.

The same problem with failure of substitutivity emerges with respect to names. Suppose that "Jane" is the name the person known as "Jill" goes by in her capacity as an undercover police detective who is trying to penetrate a network of drug traffickers. In her capacity as "Jane" she should cultivate lazy habits in order not to arouse suspicions. This is morally justified in light of the morally higher objective that she assist in penetrating the drug trafficking enclave. Now, it is true that "it should not be the case that Jill is lazy" but it is false that "it should not be the case that Jane is lazy" even though "Jill" and "Jane" refer to or denote the same individual! Extensional predicate logic does not show this sensitivity to context: if it is true in the model that "Jill is lazy" then it has to be true in the same model that "Jane is lazy" insofar as "Jill" and "Jane" so-refer. But the existence of a referentially opaque context for the deontic phrase "it should not be the case that___" shows that the logic of a deontic phrase like this is NOT extensional.

We will consider a modal logical system, appropriately called Deontic, for the charting of the logic of moral obligation and permission - and related terms. There is substantial philosophic interest in this because, in the view of the majority of philosophic logicians, this is one of the most resoundingly failed interpretations of a modal system - in other words, the formalism fails to capture the logic of deontic phrases!

## The D-System of Modal Logic

The modal system that is standardly interpreted to serve as Deontic Logic is the system called D in its syntactic construction. This is a normal system: if we approach modal systems axiomatically, we can say that normal systems have the following axioms and rules of inference (in addition to some fitting axiomatization of the propositional logic they have in them, because modal systems EXTEND standard propositional - or predicate, for that matter - logic.) [We are using, for typographic convenience, the old symbols: "L" for the strong modal operator and "M" for the weak modal operator - for instance, "necessarily" and "possibly" or, in the deontic interpretation, "obligatorily" and "permissibly." The connectives of the propositional logic extended by the modal are: {-, ->, <->, &, v}. The derivability relation is symbolized by "I-" for the propositional logic and by "II-" for the modal system.]

Axioms = {Lp <-> - M - p, L(p -> q) -> (Lp -> Lq)}

Inferential Rules = {if I- p then II- Np; if II- (N(p -> q) & Np), then II- Nq}

The semantic modeling of the matching interpretations are built up on the basis of models, M = <W, R, √>, where W is a non-empty set of possible worlds, R is an accessibility relation imposed on W, and √ is a valuation or assignment of truth values to each conceivable proposition for each possible world member of W. The semantic definitions of the modals are as follows - over the truth value set of true and false, {T, F}, indexed across the members of W. The valuations are now taken at points (possible wolds in the Kripke-Leibniz semantics.) So, √(p, wi) means "the value of the proposition denoted by "p" at possible world wi.) The accessibility relation is symbolized by "wiRwj." The possible world taken as actual in the model is "@".

√(Lp, @) = T if and only if (iff) for all wi in W, such that @Rwi, √(p, wi) = T

√(Mp, @) = T iff there is at least one wi in W, such that @Rwi and √(p, wi) = T

You can check that it is provable on these semantic definitions that the two modal operators are related in the normal way: Lp iff - M - p.

The minimal normal logic - interpreting the system constructed from the above or some other equivalent axiomatization - is: Lk, with no restrictions on the accessibility relation R. Interestingly, if R = ø - no possible world looks into any other possible world - we get exactly the case of propositional logic which appears then to be a degenerate case of modal logic. One could verify that all the axioms are provable in Lk - no countermodels can be constructed and attempt to do so runs into absurdity or contradiction.

We get additional systems by adding axioms or, for the semantic interpretations, by imposing corresponding restrictions or conditions on the relation R. The deontic axiom, D, and the corresponding condition on R, are shown below:

(D) Lp -> Mp / Condition on R: seriality: Lkser

How do we read the axiom D in the semantic interpretation. The strong modal operator in deontic logic is "it is morally obligatory that ---" and the weak modal is "it is permissible that ---." Notice that the interdefinability of the two operators (interpreting axiom 2) works, as it should: "it is morally obligatory that p iff it is not morally permissible that not-p. The other principle, interpreting Axiom 2, which gives rise to the first rule that is often called Modal Modus Ponens, also works: "if it is morally obligatory that p and if it is morally obligatory that if then q, then it is morally obligatory that q." Or does it? Perhaps it is here already that we run into difficulty. If, however, there are reservations about the naturalness of the interpretation - on account of rejecting this principle - this already poses a problem.

Now we are ready to read the deontic interpretation of the axiom D, which is added to the minimal logic to generate the system D. "If it is morally obligatory that p, then it is morally permissible that p." Most would accept this. Notice that this is not a trivial modal truth. It is true only if the model has seriality as a condition imposed on R. This means that every possible world in W has to be looking into at least one possible world (possibly itself). We should underline, however, that Lkser is not like Lkref - the reflexive model in which every world must be looking into itself. A reflexive model has to be serial (after all, every world is looking into some world - even if it is looking into itself) but a serial model is not necessarily reflexive. We must reject reflexivity for deontic matters: the characteristic axiom matching semantic reflexivity is {Np -> p}, read "if necessarily p, then actually p" for the logical necessity; for the deonic modals, this is read as "if it is morally obligatory that p, then it is actually the case that p." It is obvious that we must reject this as a deontic principle. Alas, what is morally obligatory need not be actually the case! Thus, the deontic model has no reflexivity; but it does have seriality: Lkser.

So far, so good. We are ready to see how deontic logic runs into the ground, so to speak...

## Deontic Logic and Moral Dilemmas

To anticipate what the problem is we can right away say that deontic logic is constructed in such a way that it has to take moral dilemmas as logical contradictions. There are many equivalent ways of showing this and we will try one subsequently. The crux of the matter is that a logical contradiction constitutes nonsense or absurdity. All contradictions or logical falsehoods have the same set of items that can model them: the EMPTY or null set. But moral dilemmas, on the other hand, are considered generally to be genuine - even if unfortunate - data of the moral life. So, the verdict runs, so much the worse for deontic logic itself! It doesn't get it right.

A moral dilemma arises, by definition, when a moral agent, admittedly, is morally obligated to perform (at least) two actions so that if she performs action A she cannot possibly perform action B and vice versa. So, no matter what she does in fulfilling a moral obligation, she also fails to fulfill some other moral obligation. To the question whether one should feel moral guilt about this, most answer affirmatively - although this is known to be a heatedly controversial subject in debates.

If we accepted a strong version of a principle to the effect that "ought implies can", then we can get around the difficulty. If we cannot, then we ought not! This is one way of showing how deontic logic cannot avoid scanning moral dilemmas as if they were of the logical character we attribute to contradictions. Let us see this:

1. Np & Nq (moral obligatory that the first action is performed, and the same for the second action.)
2. p <-> - q (the first can be performed if and only if the second is not.)
3. p -> - q (from 2 and given the definition of logical equivalence)
4. N(p -> - q) (from 3 and applying the rule we called Necessitation)
5. Np -> N - q (from 4, Axiom 2)
6. Np (from 1, eliminating "&")
7. N - q (6, 7 - Modus Ponens)
8. Nq (from 1, eliminating "&")
9. Nq & N-q (7, 8 - introducing "&")
10. !

Here is the crucial point. Does the expression we have in (9) capture absurdity? Let's approach it semantically. Every possible world in the deontic model sees into some possible world - because of seriality, the condition on R in a deontic model. Let's continue semantically (so, now, we have indexing of contexts or possible worlds in which propositions receive truth values; we index such worlds by {0, 1, 2, ...} starting from some actual world {0}. The accessibility relationship is symbolzied as xRy, where "x" and "y" denote possible worlds.)

1. Nq 0
2. N-q 0
3. there is some world, 1, such that 0R1 (given seriality)
4. q 1 (from 1 and 3, given the definition of deontic necessity)
5. -q 1 (from 2 and 3, given the definition of deontic necessity)
6. q & -q 1 (4, 5 - introducing "&")
7. !

We have reached logical absurdity. Remember that normal propositional modal logics extend standard propositional logic. The emergence of a logical contradiction - regardless of index - constitutes a reductio of what has been posited.

So, starting with a modal description of moral dilemma we have reached a logical absurdity.

See results

## The Demise of Deontic

We will soon see another way in which a moral dilemma is recorded as nonsense or absurdity in the formal language of deontic language. This is deemed unacceptable because a moral dilemma -- most students of the subject think -- is an ineliminable feature of our ordinary moral experience; yet, the formal system interpreted semantically for deontic purposes (to serve as a logic of moral permission and obligation) treats moral dilemmas as if they were aberrant. The absurdity that emerges when we try to capture the modal representation of a moral dilemma is not logical but is rather, itself, deontic. This sounds odd to anyone who is convinced that only one type of necessity-possibility can be understood. The alternative point of view, controversial to some, is that there are many kinds of necessity and corresponding possibility and we can talk plausibly of epistemic, physical, metaphysical - and, indeed, deontic necessity. Of course, deontic necessity is expressed formally by the strong modal operator of a deontic system. One cardinal characteristic of normal modal systems is the interdefinability between the strong and weak modal of the system: if the strong modal is symbolized as S and the weak as W, the interdefinability relationship is: Sp is logically equivalent (thus, interdefinable) with not-W-not-p. Of course the following relationships also obtain: not-Sp is equivalent with W-not-p, S-not-p is equivalent with not-Wp, If we try the semantic renderings we can easily attest that the relationship is semantically fit: necessarily-p ought to be equivalent with not-possibly-not-p, etc. For other modalities, and for the deontic in particular: obligatorily-p is equivalent with not-permissibly-not-p. The relationship between strong and weak modal operators is like that between the universal and existential quantifiers of predicate logic: all-x-are-F is equivalent with not-even-one-x-is-not-F, etc. This hints at a deep affinity between modal and predicate - but modal logics require in their metalanguages expressive resources from higher-order logics than predicate, and this plays some role in making modal logics do work that predicate logic cannot do. One could quibble as to why we should speak of deontic necessity rather than of deontic modality (reserving the term "necessity" only for logical necessity - of which, incidentally, there are more than candidate definitions.) The point is that all these modals share what see to be deep logical affinities. And this is the crux of the matter: deontic modality is inevitably treated in the same way the strongest modality - logical necessity - is to be treated. (Strength here is a matter of what can be proven. If something is provable in the strongest modal system, there can be no other system in which it is not provable.) Let us see how this, once again, makes dilemmas register as the deontic analogues of contradictions.

Suppose we symbolize the strong modality - of any modal system - by S. Our connectives are {~, ->, <->, V, &}. We are compelled to accept as a tautology of the system (with "l-" symbolizing the relation of logical consequence in our metalanguage. If we have nothing to the left of "l-", this means that what follows to the right is entailed even from the empty set of no premises at all - so, it is a tautology of the system.)

(1) l- S ~ (p & ~ p)

(1) reads: it is modally necessary that it is not the case that p and not-p. We can show this. Assuming, for reductio, that this is not the case, we have:

1. l- ~ S ~ (p & ~ p) @ -- the symbol "@" denotes the index or possible world where the value assignment is made
2. l- W (p & ~ p) @ -- this follows from (1) and given the interdefinabilty between the strong operator S and the corresponding weak operator in the system, W.
3. l- p & ~ p w1 -- the Leibniz-Kripke semantics compel evaluation of a possibilification at some other possible world, w1, which is accessible from the initial world @.
4. not-l- p & ~ p wi -- no matter what world, wi, we take, a contradiction fails to be derivable; modal systems are extensions of standard logic, for which a contradiction is always anti-designated (receives the unassertable value, which is false.)
5. not-l- p & ~ p w1 -- since w1 is one of the possible worlds, what we said about random wi world applies to w1 as well.
6. ! -- considering 3 and 5, we have an absurdity. So, since we have been doing reductio or indirect proof, we have shown: l- S ~ (p & ~ p)

The above result points to what is hiding behind the ineluctable reduction of moral dilemmas to logical absurdity. Reading the above result in deontic terms - for the deontic modal operators - we have: it is morally obligatory that not (p and not-p). Taking advantage of the relations of interdefinability between strong operator (obligatoriness) and weak operator (permissibility), we also have:

~ W (p & ~ p)

It is not permissible (deontically possible) that (p and not-p.) Dilemmas involve conflicting obligations, so that performing one entails failure to perform the other. If the propositions whose contents capture the expected act performances are "p" and "q", then we have: p iff (if and only if) not-q; or - equivalently - either p or q and not-both. The two, p and q, are related mutually by the relation known as exclusive disjunction, which we can symbolize by "+". The disequivalence (another word for exclusive disjunction) is stronger than deontic: it might even be logical necessity Q, since we still count it as a dilemma when the impossibility of dual performance is a matter of logical impossibility. We have: Q (p + q): logically necessarily, either p or q but not both. For equivalence, "<->", we have: Q ~ (p <-> q), which is equivalent with: Q (p <-> ~ q): logically necessarily, p if and only if not-q. This means that "p" are intersubstitutable in logical formulas like the one we saw above -- S ~ (p & ~ p): carrying out the substitution of "q" for "~ p", we have:

(2) l- S ~ (p & q)

This reads: it is a matter of deontic necessity that not both p and q -- for "p" and "q" standing, as we have designed them, for the two horns of a moral dilemma.

This would mean that failure to perform either p or q is not a matter of lapse in moral responsibility: it is rather a matter of not being able to do what cannot-be-done in the first place. Yet, most thinkers take moral dilemmas to present a genuine choice even notwithstanding that double performance is not possible under the circumstances. Under the result we have been tracking, there can be no genuine choice insofar as there is - presumably - no logically coherent characterization of the situation in the first place. This result vindicates - if anyone - the extreme rationalist in moral philosophy but it seems to do violence to the way moral life works in relation to moral dilemmas. Given this result, most students of philosophic ethics have opted for relinquishing the modal formalization rather than revising their views of what a moral dilemma is - they would have to consign moral dilemmas to the heap of nonsensical items (nonsensical in the relative sense of, so to speak, deontic incoherence.)

The result we have tracked could also be read as indicating that there can be no genuine moral dilemmas insofar as there are no properly intelligible moral dilemmas. This might accord with Kant's dismissal of the objection that his theory is paralyzed - cannot resolve as to how to act - in the face of moral dilemmas. Kant claimed that it is something unrelated to the proper methodology for approaching deontic phenomena that interferes and creates moral dilemmas. Kant claimed that, specifically, it is the messy empirical reality that generates dilemmas - but nothing follows from this with respect to the application of "practical reason" for the study of moral obligations. We could say, along these lines, that it is physical (or something called today metaphysical) impossibility that is responsible for moral dilemmas. From a deontic point of view, moral dilemmas are indeed nonsensical. Perhaps we should symbolize what we have in the case of a moral dilemma by using a multi-modal system that mixes deontic with physical and metaphysical modalities. (To see the distinction between physical and metaphysical modalities by means of examples, consider: "It is physically impossible for any moving item to reach or exceed the speed of light in the vacuum." "It is metaphysically impossible for anything that is not H2O to be water in any world that is like ours with respect to natural substances.") If we run our descriptions of moral dilemmas within a multi-modal system, then we should not come across any contradictions.

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